Final answer:
The magnitude of the vector ||v|| with initial point at (4,-5) and terminal point at (-1,-1) is calculated using the Pythagorean theorem, resulting in ||v|| = √(41) units.
Step-by-step explanation:
To find the exact value of the magnitude of the vector ||v||, we need to calculate the difference between the terminal and initial points of the vector in both the x and y directions.
This is done by subtracting the x-coordinate of the initial point from the x-coordinate of the terminal point, and the y-coordinate of the initial point from the y-coordinate of the terminal point.
The changes in the coordinates are as follows:
x-component: (-1) - (4) = -5
y-component: (-1) - (-5) = 4
Now we use the Pythagorean theorem to find the magnitude of the vector:
||v|| = √((-5)2 + (4)2) = √(25 + 16) = √(41)
Therefore, the exact value of the magnitude of vector v is √(41) units.